Skip to main content
SpringerLink
Log in

Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubes under moving point load

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this study, the dynamic behavior of composite beams reinforced by carbon nanotubes (CNTs) exposed to a mass moving is investigated. By considering the external potential energy due to the applied moving mass, the equations of motion of the CNT-reinforced beam are obtained using Hamilton’s principle by combining Reddy’s third-order shear deformation theory and nonlocal strain gradient theory. Three types of aligned CNT-reinforced beams are considered, namely uniformly distributed CNT beams (UD-CNT) and functionally graded CNT beams type Λ (FGΛ-CNT), and type X (FGX-CNT). Navier’s procedure is applied to obtain the closed-form solutions of simply supported CNT-reinforced beams. Verification of the present solution with previous works is presented. A detailed parametric analysis is carried out to highlight the impact of moving load velocity, nonlocal parameter, material scale parameter, total volume fraction and CNTs distribution patterns on the midspan deflections of CNTs-reinforced composite beams. The proposed model is useful in the designing and analyzing of MEMS/NEMS, nanosensor, and nanoactuator manufactured from CNTs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data availability statement

This manuscript has associated data in a data repository. [Authors’ comment: All data included in this manuscript are available upon request by contacting the corresponding author].

References

  1. A.A. Abdelrahman, N.A. Mohamed, M.A. Eltaher, Static bending of perforated nanobeams including surface energy and microstructure effects. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-01149-x

    Article  Google Scholar 

  2. A.A. Abdelrahman, M.A. Eltaher, On bending and buckling responses of perforated nanobeams including surface energy for different beams theories. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-01211-8

    Article  Google Scholar 

  3. A. Abdelrahmaan, I. Esen, C. Özarpa, M.A. Eltaher, Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory. Appl. Math. Model. 96, 215–235 (2021). https://doi.org/10.1016/j.apm.2021.03.008

    Article  MathSciNet  Google Scholar 

  4. R.M. Abo-Bakr, M.A. Eltaher, M.A. Attia, Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-01146-0

    Article  Google Scholar 

  5. L. Aichun, K. Kiani, Bilaterally flexural vibrations and instabilities of moving piezoelectric nanowires with surface effect. Eur. Phys. J. Plus 135(2), 191 (2020). https://doi.org/10.1140/epjp/s13360-020-00144-x

    Article  ADS  Google Scholar 

  6. M.S.H. Al-Furjan, R. Dehini, M. Khorami, M. Habibi, won Jung, D. , On the dynamics of the ultra-fast rotating cantilever orthotropic piezoelectric nanodisk based on nonlocal strain gradient theory. Compos. Struct. 255, 112990 (2021). https://doi.org/10.1016/j.compstruct.2020.112990

    Article  Google Scholar 

  7. R. Ansari, M.F. Shojaei, V. Mohammadi, R. Gholami, F. Sadeghi, Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams. Compos. Struct. 113, 316–327 (2014). https://doi.org/10.1016/j.compstruct.2014.03.015

    Article  Google Scholar 

  8. A. Assie, ŞD. Akbaş, A.H. Bashiri, A.A. Abdelrahman, M.A. Eltaher, Vibration response of perforated thick beam under moving load. Eur. Phys. J. Plus 136(3), 1–15 (2021). https://doi.org/10.1140/epjp/s13360-021-01224-2

    Article  Google Scholar 

  9. V. Borjalilou, E. Taati, M.T. Ahmadian, Bending, buckling and free vibration of nonlocal FG-carbon nanotube-reinforced composite nanobeams: Exact solutions. SN Appl. Sci. 1(11), 1–15 (2019). https://doi.org/10.1007/s42452-019-1359-6

    Article  Google Scholar 

  10. A.A. Daikh, A. Drai, M.S.A. Houari, M.A. Eltaher, Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes. Steel Compos. Struct. 36(6), 643–656 (2020)

    Google Scholar 

  11. A.A. Daikh, M.S.A. Houari, M.A. Eltaher, A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates. Compos. Struct. 262, 113347 (2021). https://doi.org/10.1016/j.compstruct.2020.113347

    Article  Google Scholar 

  12. F. Ebrahimi, S.H.S. Hosseini, Resonance analysis on nonlinear vibration of piezoelectric/FG porous nanocomposite subjected to moving load. Eur. Phys. J. Plus 135(2), 215 (2020). https://doi.org/10.1140/epjp/s13360-019-00011-4

    Article  Google Scholar 

  13. M. Eglin, M.A. Eriksson, R.W. Carpick, Microparticle manipulation using inertial forces. Appl. Phys. Lett. 88(9), 091913 (2006). https://doi.org/10.1063/1.2172401

    Article  ADS  Google Scholar 

  14. M.A. Eltaher, S. El-Borgi, J.N. Reddy, Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs. Compos. Struct. 153, 902–913 (2016). https://doi.org/10.1016/j.compstruct.2016.07.013

    Article  Google Scholar 

  15. M.A. Eltaher, M.A. Agwa, Analysis of size-dependent mechanical properties of CNTs mass sensor using energy equivalent model. Sens. Actuators, A 246, 9–17 (2016). https://doi.org/10.1016/j.sna.2016.05.009

    Article  Google Scholar 

  16. M.A. Eltaher, M. Agwa, A. Kabeel, Vibration analysis of material size-dependent CNTs using energy equivalent model. J. Appl. Comput. Mech. 4(2), 75–86 (2018)

    Google Scholar 

  17. M.A. Eltaher, N. Mohamed, S.A. Mohamed, Nonlinear buckling and free vibration of curved CNTs by doublet mechanics. Smart Struct. Syst. 26(2), 213–226 (2020)

    MATH  Google Scholar 

  18. M.A. Eltaher, N. Mohamed, Nonlinear stability and vibration of imperfect CNTs by doublet mechanics. Appl. Math. Comput. 382, 125311 (2020). https://doi.org/10.1016/j.amc.2020.125311

    Article  MathSciNet  MATH  Google Scholar 

  19. M.A. Eltaher, S.A. Mohamed, Buckling and stability analysis of sandwich beams subjected to varying axial loads. Steel Compos. Struct. 34(2), 241–260 (2020)

    Google Scholar 

  20. M.A. Eltaher, S.A. Mohamed, A. Melaibari, Static stability of a unified composite beams under varying axial loads. Thin-Walled Struct. 147, 106488 (2020). https://doi.org/10.1016/j.tws.2019.106488

    Article  Google Scholar 

  21. I. Esen, Dynamic response of functional graded Timoshenko beams in a thermal environment subjected to an accelerating load. Eur. J. Mech.-A/Solids 78, 103841 (2019). https://doi.org/10.1016/j.euromechsol.2019.103841

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. I. Esen, Dynamic response of a functionally graded Timoshenko beam on two-parameter elastic foundations due to a variable velocity moving mass. Int. J. Mech. Sci. 153, 21–35 (2019). https://doi.org/10.1016/j.ijmecsci.2019.01.033

    Article  Google Scholar 

  23. I. Esen, Dynamics of size-dependant Timoshenko micro beams subjected to moving loads. Int. J. Mech. Sci. 175, 105501 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105501

    Article  Google Scholar 

  24. I. Esen, A.A. Abdelrahman, M.A. Eltaher, Dynamics analysis of timoshenko perforated microbeams under moving loads. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-01212-7

    Article  Google Scholar 

  25. I. Esen, C. Özarpa, M.A. Eltaher, Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment. Compos. Struct. 261, 113552 (2021). https://doi.org/10.1016/j.compstruct.2021.113552

    Article  Google Scholar 

  26. Esen, I., Abdelrahmaan, A, Eltaher, M. A., (2021b). Free vibration and buckling stability of FG nanobeams exposed to magnetic and thermal fields. Eng. Comput.

  27. Esen, I., Eltaher, M. A., Abdelrahman, A.A., (2021c). Vibration response of symmetric and sigmoid functionally graded beam rested on elastic foundation under moving point mass. Mech. Based Des. Struct. Mach.

  28. R. Gholami, R. Ansari, Y. Gholami, Nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon-nanotube reinforced composite beams. Compos. Struct. 174, 45–58 (2017). https://doi.org/10.1016/j.compstruct.2017.04.042

    Article  Google Scholar 

  29. M. Griebel, J. Hamaekers, Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites. Comput. Methods Appl. Mech. Eng. 193(17–20), 1773–1788 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  30. M.A. Hamed, R.M. Abo-bakr, S.A. Mohamed, M.A. Eltaher, Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core. Eng. Comput. 36(4), 1929–1946 (2020). https://doi.org/10.1007/s00366-020-01023-w

    Article  Google Scholar 

  31. Y. Han, J. Elliott, Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput. Mater. Sci. 39(2), 315–323 (2007)

    Article  Google Scholar 

  32. S.H. Hashemi, H.B. Khaniki, Dynamic behavior of multi-layered viscoelastic nanobeam system embedded in a viscoelastic medium with a moving nanoparticle. J. Mech. 33(5), 559–575 (2017). https://doi.org/10.1017/jmech.2016.91

    Article  Google Scholar 

  33. S. Hashemi, H.B. Khaniki, Three dimensional dynamic response of functionally graded nanoplates under a moving load. Struct. Eng. Mech.: Int. J. 66(2), 249–262 (2018)

    Google Scholar 

  34. M. Heshmati, M.H. Yas, Free vibration analysis of functionally graded CNT-reinforced nanocomposite beam using Eshelby-Mori-Tanaka approach. J. Mech. Sci. Technol. 27(11), 3403–3408 (2013). https://doi.org/10.1007/s12206-013-0862-8

    Article  Google Scholar 

  35. M. Heshmati, M.H. Yas, F. Daneshmand, A comprehensive study on the vibrational behavior of CNT-reinforced composite beams. Compos. Struct. 125, 434–448 (2015). https://doi.org/10.1016/j.compstruct.2015.02.033

    Article  Google Scholar 

  36. L.L. Ke, J. Yang, S. Kitipornchai, Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos. Struct. 92(3), 676–683 (2010). https://doi.org/10.1016/j.compstruct.2009.09.024

    Article  Google Scholar 

  37. H.B. Khaniki, S. Hosseini-Hashemi, The size-dependent analysis of multilayered microbridge systems under a moving load/mass based on the modified couple stress theory. Eur. Phys. J. Plus 132(5), 200 (2017). https://doi.org/10.1140/epjp/i2017-11466-0

    Article  Google Scholar 

  38. H.B. Khaniki, S. Hosseini-Hashemi, Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle. Int. J. Eng. Sci. 115, 51–72 (2017). https://doi.org/10.1016/j.ijengsci.2017.02.005

    Article  MathSciNet  MATH  Google Scholar 

  39. H.B. Khaniki, On vibrations of nanobeam systems. Int. J. Eng. Sci. 124, 85–103 (2018). https://doi.org/10.1016/j.ijengsci.2017.12.010

    Article  MathSciNet  MATH  Google Scholar 

  40. H.B. Khaniki, S. Hosseini-Hashemi, H.B. Khaniki, Dynamic analysis of nano-beams embedded in a varying nonlinear elastic environment using Eringen’s two-phase local/nonlocal model. Eur. Phys. J. Plus 133(7), 1–16 (2018). https://doi.org/10.1140/epjp/i2018-12128-5

    Article  Google Scholar 

  41. H.B. Khaniki, On vibrations of FG nanobeams. Int. J. Eng. Sci. 135, 23–36 (2019). https://doi.org/10.1016/j.ijengsci.2018.11.002

    Article  MathSciNet  MATH  Google Scholar 

  42. H.B. Khaniki, M.H. Ghayesh, A review on the mechanics of carbon nanotube strengthened deformable structures. Eng. Struct. 220, 110711 (2020). https://doi.org/10.1016/j.engstruct.2020.110711

    Article  Google Scholar 

  43. H.B. Khaniki, M.H. Ghayesh, On the dynamics of axially functionally graded CNT strengthened deformable beams. Eur. Phys. J. Plus 135(5), 415 (2020). https://doi.org/10.1140/epjp/s13360-020-00433-5

    Article  Google Scholar 

  44. H.B. Khaniki, M.H. Ghayesh, S. Hussain, M. Amabili, Porosity, mass and geometric imperfection sensitivity in coupled vibration characteristics of CNT-strengthened beams with different boundary conditions. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-01208-3

    Article  Google Scholar 

  45. L. Li, Y. Hu, Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 107, 77–97 (2016). https://doi.org/10.1016/j.ijengsci.2016.02.010

    Article  MathSciNet  MATH  Google Scholar 

  46. C.W. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015). https://doi.org/10.1016/j.jmps.2015.02.001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. F. Lin, Y. Xiang, Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Appl. Math. Model. 38(15–16), 3741–3754 (2014). https://doi.org/10.1016/j.apm.2014.02.008

    Article  MathSciNet  MATH  Google Scholar 

  48. H. Liu, Q. Zhang, J. Ma, Thermo-mechanical dynamics of two-dimensional FG microbeam subjected to a moving harmonic load. Acta Astronaut. 178, 681–692 (2021). https://doi.org/10.1016/j.actaastro.2020.09.045

    Article  ADS  Google Scholar 

  49. S.N. Mahmoodi, S.E. Khadem, N. Jalili, Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam. Arch. Appl. Mech. 75(2), 153–163 (2006). https://doi.org/10.1007/s00419-005-0426-1

    Article  ADS  MATH  Google Scholar 

  50. A. Melaibari, A.B. Khoshaim, S.A. Mohamed, M.A. Eltaher, Static stability and of symmetric and sigmoid functionally graded beam under variable axial load. Steel Compos. Struct. 35(5), 671–685 (2020)

    Google Scholar 

  51. S.S. Mirjavadi, M. Forsat, M.R. Barati, G.M. Abdella, B.M. Afshari, A.M.S. Hamouda, S. Rabby, Dynamic response of metal foam FG porous cylindrical micro-shells due to moving loads with strain gradient size-dependency. Eur. Phys. J. Plus 134(5), 1–11 (2019)

    Google Scholar 

  52. N. Mohamed, S.A. Mohamed, M.A. Eltaher, Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-00976-2

    Article  Google Scholar 

  53. C. Özarpa, I. Esen, Modelling the dynamics of a nanocapillary system with a moving mass using the non-local strain gradient theory. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6812

    Article  Google Scholar 

  54. M. Rafiee, J. Yang, S. Kitipornchai, Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers. Compos. Struct. 96, 716–725 (2013). https://doi.org/10.1016/j.compstruct.2012.10.005

    Article  Google Scholar 

  55. O. Rahmani, M. Shokrnia, H. Golmohammadi, S.A.H. Hosseini, Dynamic response of a single-walled carbon nanotube under a moving harmonic load by considering modified nonlocal elasticity theory. Eur. Phys. J. Plus 133(2), 1–13 (2018). https://doi.org/10.1140/epjp/i2018-11868-4

    Article  Google Scholar 

  56. S. Rajasekaran, H.B. Khaniki, Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass. Appl. Math. Model. 72, 129–154 (2019). https://doi.org/10.1016/j.apm.2019.03.021

    Article  MathSciNet  MATH  Google Scholar 

  57. J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2–8), 288–307 (2007). https://doi.org/10.1016/j.ijengsci.2007.04.004

    Article  MATH  Google Scholar 

  58. M.A. Roudbari, T.D. Jorshari, A.G. Arani, C. Lü, T. Rabczuk, Transient responses of two mutually interacting single-walled boron nitride nanotubes induced by a moving nanoparticle. Eur. J. Mech.-A/Solids 82, 103978 (2020). https://doi.org/10.1016/j.euromechsol.2020.103978

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. She, G. L., Liu, H. B., & Karami, B. Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets. Thin-Walled Struct. 160:107407.

  60. H.S. Shen, Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Compos. Struct. 91(1), 9–19 (2009). https://doi.org/10.1016/j.compstruct.2009.04.026

    Article  Google Scholar 

  61. H.S. Shen, Y. Xiang, Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Comput. Methods Appl. Mech. Eng. 213, 196–205 (2012). https://doi.org/10.1016/j.cma.2011.11.025

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. A.G. Shenas, P. Malekzadeh, S. Ziaee, Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment. Compos. Struct. 162, 325–340 (2017). https://doi.org/10.1016/j.compstruct.2016.12.009

    Article  Google Scholar 

  63. M. Subramani, S. Rajeshkumar, M. Ramamoorthy, Free vibration analysis of the MWCNT reinforced hybrid laminated composite sandwich beam. Mater. Today: Proc. 22, 3220–3225 (2020). https://doi.org/10.1016/j.matpr.2020.03.460

    Article  Google Scholar 

  64. E. Taati, V. Borjalilou, Fallah, and F., & Ahmadian, M. T. , On size-dependent nonlinear free vibration of carbon nanotube-reinforced beams based on the nonlocal elasticity theory: perturbation technique. Mech. Based Des. Struct. Mach. (2020). https://doi.org/10.1080/15397734.2020.1772087

    Article  Google Scholar 

  65. N. Wattanasakulpong, V. Ungbhakorn, Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Comput. Mater. Sci. 71, 201–208 (2013). https://doi.org/10.1016/j.commatsci.2013.01.028

    Article  Google Scholar 

  66. H.L. Wu, J. Yang, S. Kitipornchai, Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections. Compos. B Eng. 90, 86–96 (2016). https://doi.org/10.1016/j.compositesb.2015.12.007

    Article  Google Scholar 

  67. Z. Wu, Y. Zhang, G. Yao, Z. Yang, Nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite beams. Int. J. Mech. Sci. 153, 321–340 (2019). https://doi.org/10.1016/j.ijmecsci.2019.02.015

    Article  Google Scholar 

  68. M.H. Yas, M. Heshmati, Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load. Appl. Math. Model. 36(4), 1371–1394 (2012). https://doi.org/10.1016/j.apm.2011.08.037

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Karabuk University, Karabuk, Turkey

    Ismail Esen

  2. Department of Civil Engineering, Laboratoire D’Etude Des Structures Et de Méecanique Des Matéeriaux, Mascara, Algeria

    Ahmed Amin Daikh

  3. Mechanics of Structures and Solids Laboratory, Faculty of Technology, University of Sidi Bel Abbes, Sidi Bel Abbes, Algeria

    Ahmed Amin Daikh

  4. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    Mohamed A. Eltaher

  5. Mechanical Design and Production Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt

    Mohamed A. Eltaher

Authors
  1. Ismail Esen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed Amin Daikh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed A. Eltaher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed A. Eltaher.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Esen, I., Daikh, A.A. & Eltaher, M.A. Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubes under moving point load. Eur. Phys. J. Plus 136, 458 (2021). https://doi.org/10.1140/epjp/s13360-021-01419-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01419-7

Search

Navigation